President Obama and Mitt Romney revived their high-decibel complaints about China during Tuesday’s debate, each seeking to impress voters with how tough they would be on a country that for decades has served presidential candidates as a convenient villain.

“I’ll crack down on China when they cheat,” Romney pledged, citing China’s record on currency and trade that results in a loss of American jobs and patented technology.

Obama responded with a boast of his own: “We’ve put unprecedented trade pressure on China” and he repeatedly cited Romney’s investments in China through the private equity firm he founded, Bain Capital.

The rhetoric has obvious appeal to voters uneasy with the general state of the economy and resonates particularly in battleground industrial states like Ohio, Wisconsin and Michigan, where residents still feel a visceral impact from the decline of manufacturing jobs. But Obama and Romney know considerably more about the complex realities of dealing with China than they let on — and neither candidate approached fundamental questions raised for the U.S. by the rise of China as an economic power.

Arguably, Romney has the most sophisticated understanding of the global economy of any recent presidential candidate. He founded Bain Capital by drawing international investors to a partnership that produced extraordinary profits anticipating global market trends, including the outsourcing of jobs once done in-house by American manufacturers.

Anticipating attacks on this business record, Romney appears reticent to publicly discuss his considerable international investment experience. In part, that’s because he understands how the word “China” can be used to motivate worried voters.

Bill Clinton deployed the China card in 1992, complaining that his Republican predecessor had coddled the “butchers of Beijing.” Eight years later, George W. Bush criticized Clinton for being too easy on China. And Romney saw Obama use China during the 2008 debate, with his own vow to crack down on currency and trade manipulation, citing President Bush’s failure to take action.

During Tuesday’s debate Romney pledged repeatedly that he would move swiftly against China’s alleged scheme to artificially hold down the price of its currency, the yuan. A low value for the yuan makes Chinese exports cheaper around the world.

“On day one, I will label China a currency manipulator, which will allow me as president to be able to put in place, if necessary, tariffs where I believe that they are taking unfair advantage of our manufacturers,” Romney said.

Obama made similar promises as a candidate in 2008, but now cites the advantage of informal jawboning with the Chinese. “The currency has actually gone up 11 percent since I’ve been president because we have pushed them hard,” he said during the debate.

Despite the attention it received, the pricing of the yuan is a sideshow to the fundamental questions about America’s relationship with China, which now owns a large portion of U.S. debt and is inextricably tied to the overall economy.

Candy Crowley, the tough-minded moderator, asked the candidates a penetrating question about U.S.-China relations. “The iPad, the Macs, the iPhones, they are all manufactured in China,” she said. “One of the major reasons is labor is so much cheaper” in China. “How do you convince a great American company to bring that manufacturing back here?”

Neither candidate dealt fully with her question, which gets to the fundamental dynamic of the 21st century globalized economy: Jobs are streaming to low-wage countries, providing advantages to U.S. manufacturers like Apple, but also for American consumers who can obtain cheaper goods.

Missing from the discussion was the frank testimony of those who watched manufacturing jobs disappear and understand why they go. Andrew Grove, former CEO of Intel, argued in 2010 that American needs to reconsider its approach to outsourcing and globalization.

The trend, in which his former company has vigorously participated, is starting to cost the country in its technical and economic prowess, he wrote. “You could say, as many do, that shipping jobs overseas is no big deal because the high-value work — and much of the profits — remain in the U.S.,” he wrote in an essay published in 2010 by Business Week. “But what kind of a society are we going to have if it consists of highly paid people doing high-value-added work — and masses of unemployed?”

Obama declared that Mitt Romney was “the last person who is going get tough with China.” He cited Romney investments in Chinese companies, some of which were “pioneers” in outsourcing and in a firm that engaged in surveillance of Chinese citizens.

Romney acknowledged holding some investments in China, but pointed out that these have been managed by a blind trust for the past eight years. He then made his own important point that went unanswered: He said Obama also has “investments in China.”

Bain Capital, the private equity firm Romney founded, did invest in firms that specialized in outsourcing during the years Romney ran the private equity firm.

In using the term “pioneer,” Obama was referring to a story in The Washington Post that detailed early investments made by Bain in companies that would become major players in outsourcing to China and other countries.

In addition, Romney has a stake in a Bain investment fund that owns Uniview Technologies, a Chinese firm that produces infrared cameras and surveillance gear. That investment did not occur until late 2011, long after Romney left Bain Capital.

Obama deflected questions about his own investments in China with a joke. But Obama holds shares in mutual funds that invest in companies like Apple with operations in China.

The ubiquitous nature of investment in China is not a joke.These investments are a fundamental part of the U.S. economy now. That explains something Obama and Romney both know but won’t say: Dealing with China and the issue of outsourcing is no simple matter.